Analysis of the Double Scattering Scintillation of Waves in Random Media

Abstract

High frequency waves propagating in highly oscillatory media are often modeled by radiative transfer equations that describes the propagation of the energy density of the waves. When the medium is statistically homogeneous, averaging effects occur in such a way that in the limit of vanishing wavelength, the wave energy density solves a deterministic radiative transfer equation. In this paper, we are interested in the remaining stochasticity of the energy density. More precisely, we wish to understand how such stochasticity depends on the statistics of the random medium and on the initial phase-space structure of the propagating wave packets. The analysis of stochasticity is a formidable task involving complicated analytical calculations. In this paper, we consider the propagation of waves modeled by a scalar Schrödinger equation and limit the interaction of the waves with the underlying structure to second order. We calculate the scintillation function (second statistical moment of the Wigner transform) for such signals, which thus involve fourth-order moments of the random fluctuations, which we assume have Gaussian statistics. Our main result is a detailed analysis of the scintillation function in that setting. This requires the analysis of non-trivial oscillatory integrals, which is carried out in detail.